THE STUDY OF NON-LINEAR PHENOMENA BY THE ASYMPTOTIC ANALYSIS

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The purpose of this project is to describe nonlinear phenomena mathematically by using asymptotic analysis. And we have the following results.1) Narukawa and Fukagai proposed a mathematical model related to the nonlinear elasticity. This is described by a degenerate quasilinear elliptic equation whose principal part has different orders at 0 and at infinity. They have showed a global bifurcation diagram of positive solutions for a nonlinear eigenvalue problem of such quasilinear equations and, in particular, the coexistence of multiple positive solutions. These results obtained by regularity estimate of weak solutions and modifying the argument given by Ambrosetti, Brezis and Cerami in the semilinear case.2) Ohnuma has investigated a class of singular degenerate parabolic equations including the p-Laplace diffusion equation and the equation of the mean curvature flow, and proved the comparison principle for these equations. He also discovered a strong maximum principle of quasilinear degenerate elliptic equations.3) Murakami showed a necessary and sufficient condition of the asymptotic stability of a fixed point for a higher order linear difference equation. He has also investigated some nonlinear difference equations, derived the formula to compute the stability conditions of the invariant curve caused by the Neimark-Sacker bifurcation and, moreover, given the explicit expression of the invariant curve.4) Kohda has obtained blow-up criteria for a solution of an initial value problem of a semilinear parabolic equation. This is described by using a super-solusion and a sub-solution of the stationary problem. Moreover, he had given some condition which guarantees the blow-up of the solution.